Vibration localization and wave conversion phenomena in a multi-coupled, nearly periodic, disordered truss beam

An investigation of the dynamics of periodic and nearly periodic disordered truss beams is presented. An exact wave transfer matrix methodology is chosen to describe the characteristic harmonic waves and the normal modes of vibration of these multi-coupled periodic structures. The method is found to capture accurately the dynamics of truss beams in wide frequency ranges, even at those higher frequencies where the individual truss members undergo resonance and where the modal density is high. The methodology is extended to examine the effects of slight disorder among the bays upon the propagation of waves and the transmission of vibration through the truss beam. However, the approach faces severe numerical problems for the disordered system when taking the product of random wave transfer matrices. In order to circumvent this difficulty, a new algorithm is developed to calculate accurately the wave amplitudes at all bays. It is shown that both free harmonic waves and modes of vibration that extend throughout the entire ordered structure may become localized to a few of the bays of the disordered truss beam. The concept of power flow is utilized as a scalar descriptor of localization to examine the interactions among the various waves present in the structure. Results show that leakage of energy occurs from the wave pair that is most subject to localization to wave pairs that are less prone to localization, suggesting that localization is more difficult to achieve in multi-coupled systems than in mono-coupled ones. The wave conversion mechanism observed here is a novel phenomenon that is characteristic of disorder effects in multi-coupled structures. Lyapunov exponents are applied to examine the asymptotic decay rates of waves in both mono- and multi-coupled systems. It is found that the decay rate of each wave tends to decrease from the largest Lyapunov exponent to the smallest one. When the wave conversion is significant, energy tends to transmit along the system according to the smallest Lyapunov exponent.